Spectral extremal results on edge blow-up of graphs

Abstract

Let $\mathrm{ex}(n,F)$ and $\mathrm{spex}(n,F)$ be the maximum size and the maximum spectral radius of an F-free graph of order n, respectively. The value $\mathrm{spex}(n,F)$ is called the spectral extremal value of F. Nikiforov (2009) [24] gave the spectral Stability Lemma, which implies that for every $\epsilon >0$, sufficiently large n and a non-bipartite graph H with chromatic number $\chi \left(H\right)$, the extremal graph for $\mathrm{spex}(n,H)$ can be obtained from the Turán graph $T_{\chi \left(H\right)-1}\left(n\right)$ by adding and deleting at most $\epsilon n^2$ edges. It is still a challenging problem to determine the exact spectral extremal values of many non-bipartite graphs. Given a graph F and an integer $p\ge 2$, the edge blow-up of F, denoted by $F^{p+1}$, is the graph obtained from replacing each edge in F by a $K_{p+1}$ where the new vertices of $K_{p+1}$ are all distinct. In this paper, we determine the exact spectral extremal values of the edge blow-up of all non-bipartite graphs and provide the asymptotic spectral extremal values of the edge blow-up of all bipartite graphs for sufficiently large n, which can be seen as a spectral version of the theorem on $\mathrm{ex}(n,F^{p+1})$ given by Yuan (2022) [34]. As applications, on the one hand, we generalize several previous results on $\mathrm{spex}(n,F^{p+1})$ for F being a matching and a star. On the other hand, we obtain the exact values of $\mathrm{spex}(n,F^{p+1})$ for F being a path, a cycle, and a complete graph.